Conditional Diagnosability of the BC Networks under
the Comparison Diagnosis Model
*+Guo-Huang Hsu
a, Jimmy J. M. Tan
bDepartment of Computer Science, National Chiao Tung University,
Hsinchu, Taiwan 300, R.O.C.
a
[email protected],
b[email protected]
Abstract-
An n-dimensional bijective connection network (BC network), denoted by Xn, is an n-regular graph with 2n vertices and n2n-1 edges. The n-dimensional hypercube, crossed cube, twisted cube, and Möbius cube are some examples of the n-dimensional BC networks. In [5], Lai et al. introduced a novel measure of diagnosability, called conditional diagnosability, by adding an additional condition that any faulty set cannot contain all the neighbors of any vertex in a system. In this paper, we prove that the conditional diagnosability of Xn is 3(n-2)+1 under the comparison model, n≥5. As a corollary of this result, we obtain the conditional diagnosability of the hypercubes, crossed cubes, twisted cubes, and Möbius cubes.Keywords: comparison diagnosis model, diagnosability, conditional diagnosability, BC network.
1. Introduction
The problem of fault diagnosis in multiprocessor systems has gained increasing importance and has been widely studied in the literatures [2], [3], [5], [6], [11], [13]. In order to diagnose a multiprocessor system, several different models have been proposed [7], [9]. Throughout this paper, we base our diagnosability analysis on the comparison model. The comparison model deals with the faulty diagnosis by sending the same input (or task) from a vertex w to each pair of distinct neighbors, u and v, and then comparing their responses. The vertex w is called the comparator of vertices u and v. The result of the comparison is either the two responses agreed or two responses disagreed. Based on the results of all the comparisons, the system can decide the faulty or fault-free status of the vertices.
Reviewing some previous papers [2], [3], [6], [11], the Hypercube Qn, the Crossed cube CQn, the Twisted
cube TQn, and the Möbius cube MQn, all have
diagnosability n under the comparison model. In classical
measures of system-level diagnosability for multiprocessor systems, if all the neighbors of some processor v are faulty simultaneously, it is not possible to determine whether processor v is fault-free or faulty. As a consequence, the diagnosability of a system is limited by its minimum degree. Therefore, Lai et al. introduced a restricted diagnosability of multiprocessor systems called conditional diagnosability in [5]. Lai et al. considered a measure by restricting that, for each processor v in a system, not all the processors which are directly connected to v fail at the same time. In this paper, we prove that the conditional diagnosability of
n-dimensional BC networks Xn is 3(n-2)+1 under the
comparison model, n≥5. As a corollary of this result, we obtain the conditional diagnosability of the hypercubes, crossed cubes, twisted cubes, and Möbius cubes.
2. Preliminaries
For the graph definition and notation we follow [12]. A multiprocessor system can be modeled as a graph
G(V,E), where the set of vertices V represents processors
and the set of edges E represents communication links between processors.
Let G(V,E) be a graph and v∈V(G) be a vertex. The neighborhood N(v) of vertex v is the set of all vertices that are adjacent to v. The cardinality |N(v)| is called the degree of v, denoted by degG(v) or simply deg(v). For a
subset of vertices V'⊂V(G), the neighborhood set of the vertex set V' is defined as N(V')=
' ( ) '. v V N v V ∈ −
∪
For a setof vertices(respectively, edges) S, we use the notation
G – S to denote the graph obtained from G by removing
all the vertices(respectively, edges) in S. The components of a graph G are its maximal connected subgraphs. A component is trivial if it has no edges; otherwise, it is nontrivial. The connectivity κ(G) of a graph G(V,E) is the minimum number of vertices whose removal results in a disconnected or a trivial graph. Let F1,F2⊆V(G) be two
distinct sets. The symmetric difference of the two sets F1
and F2 is defined as the set F1ΔF2 = (F1 – F2)∪(F2 – F1).
The comparison model[7] is proposed by Malek and Maeng. In this model, a self-diagnosable system is often represented by a multigraph M(V,C), where V is the same vertex set defined in G and C is the labeled edge set. Let *This research was partially supported by the National
Science Council of the Republic of China under contract NSC 95-2221-E-009-134-MY3.
+This research was partially supported by the Aiming for the Top University and Elite Research Center Development Plan.
(u,v)w be a labeled edge. If (u,v) is an edge labeled by w,
then (u,v)w is said to belong to C, which implies that the
vertex u and v are being compared by vertex w. The same pair of vertices may be compared by different comparators, so M is a multigraph. For (u,v)w∈C, we use r((u,v)w) to denote the result of comparing vertices u and v by w such that r((u,v)w)=0 if the outputs of u and v agree,
and r((u,v)w)=1 if the outputs disagree. In this model, if r((u,v)w)=0 and w is fault-free, then both u and v are
fault-free. If r((u,v)w)=1, then at least one of the three
vertices u, v, w must be faulty. If the comparator w is faulty, then the result of comparison is unreliable that means both r((u,v)w)=0 and r((u,v)w)=1 are possible
outputs, and it outputs only one of these two possibilities. The collection of all comparison results, defined as a function σ: C→{0,1}, is called the syndrome of the diagnosis. A subset F⊂V is said to be compatible with a syndrome σ if σ can arise from the circumstance that all vertices in F are faulty and all vertices in V–F are fault-free. A system is said to be diagnosable if, for every syndrome σ, there is a unique F⊂V that is compatible with σ. In [10], a system is called a t-diagnosable system if the system is diagnosable as long as the number of faulty vertices does not exceed t. The maximum number of faulty vertices that the system G can guarantee to identify is called the diagnosability of G, written as t(G). Let σF={σ | σ is compatible with F}. Two distinct sets F1,F2⊂V are said to be indistinguishable if and only if
σF1∩σF2≠∅; otherwise, F1,F2 are said to be
distinguishable. The following theorem given by
Sengupta and Dahbura [10] is a necessary and sufficient condition for ensuring distinguishability.
Theorem 1. [10] Let G(V,E) be a graph. For any two distinct sets F1,F2⊂V, (F1,F2) is a distinguishable pair if
and only if at least one of the following conditions is satisfied (see Figure 1):
1. ∃u,w∈V–F1–F2 and ∃v∈F1ΔF2 such that (u,v)w∈C,
2. ∃u,v∈F1–F2 and ∃w∈V–F1–F2 such that (u,v)w∈C,
or
3. ∃u,v∈F2–F1 and ∃w∈V–F1–F2 such that (u,v)w∈C
Figure 1: Description of distinguishability for Theorem 1.
An n-dimensional bijective connection network (BC network), denoted by Xn, is an n-regular graph with 2n
vertices and n2n-1 edges. The set of all the n-dimensional BC networks is called the family of the n-dimensional
BC networks, denoted by Ln. Xn and Ln may be
recursively defined as below [4].
Definition 1. The 1-dimensional BC graph X1 is a
complete graph with two vertices. The family of the 1-dimensional BC graph is defined as L1 = {X1}. Let G
be a graph. G is an n-dimensional BC graph, denoted by
Xn, if there exist V0, V1⊂V(G) such that the following
two conditions hold:
1. V(G) = V0∪V1, V0≠∅, V1≠∅, V0∩V1=∅; and
2. There exists an edge set M⊂E(G) such that M is a perfect matching between V0 and V1, G(V0)∈Ln-1
and G(V1)∈Ln-1.
Before studying the conditional diagnosability of the BC networks, we need some definitions for further discussion. Let G(V,E) be a graph. For any set of vertices U⊆V(G), G[U] denotes the subgraph of G induced by the vertex subset U. Let H be a subgraph of
G and v be a vertex in H. We use V(H;3)={v∈V(H) | degH(v)≥3} to represent the set of vertices which has
degree 3 or more in H. Let F1,F2⊆V(G) be two distinct
sets and S=F1∩F2. We use CF1ΔF2,S to denote the
subgraph induced by the vertex subset (F1ΔF2)∪{u |
there exists a vertex v∈ F1ΔF2 such that u and v are
connected in G–S}. The following result is a useful sufficient condition for checking whether (F1,F2) is a
distinguishable pair.
Theorem 2. Let G(V,E) be a graph. For any two distinct sets F1,F2⊂V with |Fi| ≤ t, i=1,2, and S=F1∩F2. (F1,F2) is
distinguishable if, the subgraph CF1ΔF2,S of G–S contains
at least 2(t-|S|)+1 vertices having degree 3 or more. Proof.
Given any pair of distinct sets of vertices F1,F2⊂V
with |Fi| ≤ t, i=1,2. Let S=F1∩F2, then 0 ≤ |S| ≤ t–1, and
|F1ΔF2| ≤ 2(t–|S|). Consider the subgraph CF1ΔF2,S, the
number of vertices having degree 3 or more is at least 2(t-|S|)+1 in CF1ΔF2,S, the subgraph CF1ΔF2,S contains at
least 2(t-|S|)+1 vertices. There is at least one vertex with degree 3 or more lying in CF1ΔF2,S–F1ΔF2. Let u be one
of such vertices with degree 3 or more. Let i, j, and k be three distinct vertices linked to u. If one of i, j, and k lies in CF1ΔF2,S–F1ΔF2, condition 1 of Theorem 1 holds
obviously. Suppose all these three vertices belong to
F1ΔF2. Without loss of generality, assume i lies in F1-F2,
one of the two cases will happen: 1) if j lies in F1-F2,
condition 2 of Theorem 1 holds; or, 2) if j lies in F2-F1,
wherever k lies in F1-F2 or F2-F1, condition 2 or 3 of
Theorem 1 holds. So (F1,F2) is a distinguishable pair
and the proof is complete. By Theorem 2, we now propose a sufficient condition to verify whether a system is t-diagnosable under the comparison model.
Corollary 1. Let G(V,E) be a graph. G is t-diagnosable if, for each set of vertices S⊂V with |S| = p, 0≤ p ≤ t-1, every connected component C of G–S contains at least 2(t-p)+1 vertices having degree at least three. More precisely, |V(C;3)| ≥ 2(t-p)+1.
3. Conditional Diagnosability of BC
Networks X
nIn classical measures of diagnosability for multiprocessor systems under the comparison model, if all the neighbors of some processor v are faulty simultaneously, it is not possible to determine whether processor v is fault-free or faulty. So the diagnosability of a system is limited by its minimum vertex degree.
In an n-dimensional Hypercube Qn, Qn has 2
n
n ⎛ ⎞ ⎜ ⎟ ⎝ ⎠
vertex subsets of size n, among which there are only 2n vertex subsets which contains all the neighbors of some vertex. Since the ratio 2n/ 2n
n ⎛ ⎞ ⎜ ⎟ ⎝ ⎠
is very small for large n, the probability of a faulty set containing all the neighbors of any vertex is very low. For this reason, Lai et al. introduced a new restricted diagnosability of multiprocessor systems called conditional diagnosability in [5]. They consider the situation that any faulty set cannot contain all the neighbors of any vertex in a system. In the following, we need some terms to define the conditional diagnosability formally. A faulty set F⊂V is called a conditional faulty set if N(v)⊄F for every vertex
v∈V. A system G(V,E) is said to be conditionally t-diagnosable if F1 and F2 are distinguishable, for each
pair of conditional faulty sets F1,F2⊂V, and F1≠F2, with
|F1| ≤ t and |F2| ≤ t. The maximum value of t such that G
is conditionally t-diagnosable is called the conditional
diagnosability of G, written as tc(G). It is trivial that tc(G) ≥ t(G).
Lemma 1. Let G be a multiprocessor system. Then,
tc(G) ≥ t(G).
Now, we give an example to show that the conditional diagnosability of the BC graph Xn is no
greater than 3(n-2)+2, n ≥ 5. As shown in Figure 2, we take a cycle of length four in Xn. Let {v1,v2,v3,v4} be the
four consecutive vertices on this cycle, and let
F1=N({v1,v3,v4})∪{v1} and F2=N({v1,v3,v4})∪{v3}, then
|F1|=|F2|=3(n-2)+2. It is straightforward to check that F1
and F2 are two conditional faulty sets, and F1 and F2 are
indistinguishable by Theorem 1. Note that the BC graph
Xn has no cycle of length 3 and any two vertices have at
most two common neighbors. As we can see, |F1-F2|=|F2-F1|=1 and |F1∩F2|=3(n-2)+1. Therefore, Xn is
not conditionally (3(n-2)+2)-diagnosable and
tc(Xn)≤3(n-2)+1, n ≥ 3. Then, we shall show that Xn is
conditionally t-diagnosable, where t=3(n-2)+1. Lemma 2. tc(Xn) ≤ 3(n-2)+1 for n ≥ 3.
Figure 2: An indistinguishable conditional-pair (F1,F2),
where |F1|=|F2|=3(n-2)+2.
Let F be a set of vertices F⊂V(Xn) and C be a
connected component of Xn-F. We need some results on
the cardinalities of F and V(C) under some restricted conditions. The results are listed in Lemma 3 and 4. In Lemma 3, Zhu proved that deleting at most 2(n-1)-1 vertices from Xn, the incomplete BC graph Xn has one
connected component containing at least 2n-|F|-1 vertices. We expand this result further. In Lemma 4, we show that deleting at most 3n-6 vertices from Xn, the
incomplete BC graph Xn has one connected component
containing at least 2n-|F|-2 vertices.
Lemma 3. [14] ∀Xn∈Ln (n ≥ 3), let F be a set of vertices F⊂V(Xn) with n ≤ |F| ≤ 2(n-1)-1. Suppose that Xn-F is
disconnected. Then Xn-F has exactly two components,
one is trivial and the other is nontrivial. The nontrivial component of Xn-F contains 2n-|F|-1 vertices.
The BC graph can be described as follows: Let Xn
denote an n-dimensional BC graph. X1 is a complete
graph with two vertices labeled with 0 and 1, respectively. For n ≥ 2, each Xn consists of two Xn-1's,
denoted by Xn-1L and Xn-1R, with a perfect matching M
between them. That is, M is a set of edges connecting the vertices of Xn-1L and the vertices of Xn-1R in a
one-to-one manner. It is easy to see that there are 2n-1 edges between Xn-1L and Xn-1R. By using a simple
induction, we can prove the following lemma.
Lemma 4. ∀Xn∈Ln (n ≥ 5), let F be a set of vertices F⊂V(Xn) with |F| ≤ 3n-6. Then Xn-F has a connected
component containing at least 2n-|F|-2 vertices. Proof.
We prove the lemma by induction on n. For n = 5, it is straightforward to verify that the lemma holds. As the inductive hypothesis, we assume that the result is true for Xn-1, for |F| ≤ 3(n-1)-6, and for some n ≥ 6. Now we
consider Xn, |F| ≤ 3n-6. An n-dimensional BC graph Xn
can be divided into two Xn-1's, denoted by Xn-1L and Xn-1R.
Let FL = F∩V(Xn-1L), 0 ≤ |FL| ≤ 3n-6 and FR = F∩V(Xn-1R), 0 ≤ |FR| ≤ 3n-6. Then |F| = |FL| + |FR|.
Without loss of generality, we may assume that |FL|≥|FR|.
In the following proof, we consider two cases by the size of FR: 1) 0 ≤ |FR| ≤ 2 and 2) |FR| ≥ 3.
Since 0 ≤ |FR| ≤ 2, Xn-1R-FR is connected and
|V(Xn-1R-FR)|=2n-1-|FR|. Let FR(L)⊂V(Xn-1L) be the set of
vertices which has neighboring vertices in FR. For each
vertex v∈Xn-1L-FL-FR(L), there is exactly one vertex v(R)
in Xn-1R-FR, such that (v,v(R))∈E(Xn). Besides,
|V(Xn-1L-FL-FR(L))| ≥ 2n-1-|FL|-|FR|. Hence Xn-F has a
connected component that contains at least [2n-1-|FR|] +
[2n-1-|FL|-|FR|] = 2n-|F|-|FR| ≥ 2n-|F|-2 vertices.
Case 2: |FR| ≥ 3.
Since |FR| ≥ 3, 3 ≤ |FL| ≤ 3(n-1)-6 and 3 ≤ |FR| ≤
3(n-1)-6. By the inductive hypothesis, Xn-1L-FL (Xn-1R-FR,
respectively) has a connected component CL (CR,
respectively) that contains at least 2n-1-|FL|-2 (2n-1-|FR|-2,
respectively) vertices. Next, we divide the case into three subcases: 2.1) |V(CL)|=2n-1-|FL|-2 and Xn-1R-FR is
disconnected, 2.2) |V(CL)|=2n-1-|FL|-2 and Xn-1R-FR is
connected, and 2.3) |V(CL)| ≥ 2n-1-|FL|-1 and |V(CR)| ≥
2n-1-|FR|-1.
Case2.1: |V(CL)|=2n-1-|FL|-2 and Xn-1R-FR is disconnected
This is an impossible case. Since κ(Xn-1)=n-1, |FR| ≥ n-1. By Lemma 3, |FL| ≥ 2((n-1)-1). Then the total
number of faulty vertices is at least (n-1) + 2((n-1)-1) = 3n-5 which is greater than 3n-6, a contradiction.
Case 2.2: |V(CL)|=2n-1-|FL|-2 and Xn-1R-FR is connected.
Since Xn-1R-FR is connected, |V(Xn-1R-FR)| = 2n-1-|FR|.
Since |V(CL)| ≥ |FR| + 1, there exists a vertex u∈CL and a
vertex v∈CR such that (u,v)∈ E(Xn). Hence Xn-F has a
connected component that contains at least [2n-1-|FR|] +
[2n-1-|FL|-2] = 2n-|F|-2 vertices.
Case 2.3: |V(CL)| ≥ 2n-1-|FL|-1 and |V(CR)| ≥ 2n-1-|FR|-1.
Since |V(CL)| ≥ |FR| + 1, there exists a vertex u∈CL
and a vertex v∈CR such that (u,v) ∈ E(Xn). Hence Xn-F
has a connected component that contains at least [2n-1-|FL|-1] + [2n-1-|FR|-1] = 2n-|F|-2 vertices.
This completes the proof of the lemma. By Lemma 4, we have the following corollary. Corollary 2. ∀Xn∈Ln (n ≥ 5), let F be a set of vertices F⊂V(Xn) with |F| ≤ 3n-6. Then Xn-F satisfies one of the
following conditions: 1. Xn-F is connected.
2. Xn-F has two components, one of which is K1, and
the other one has 2n-|F|-1 vertices.
3. Xn-F has two components, one of which is K2, and
the other one has 2n-|F|-2 vertices.
4. Xn-F has three components, two of which are K1,
and the third one has 2n-|F|-2 vertices.
We are now ready to show that the conditional diagnosability of Xn is 3(n–2)+1 for n ≥ 5. Let F1,F2⊂V(Xn) be two conditional faulty sets with F1 ≤
3(n-2)+1 and F2 ≤ 3(n-2)+1, n ≥ 5. We shall show our
result by proving that (F1,F2) is a distinguishable
conditional-pair under the comparison model.
Lemma 6. Let Xn be an n-dimensional BC graph with n≥5. For any two conditional faulty sets F1,F2⊂V(Xn),
and F1 ≠ F2, with F1 ≤ 3(n-2)+1 and F2 ≤ 3(n-2)+1. Then
(F1,F2) is a distinguishable conditional-pair under the
comparison model. Proof.
We use Theorem 2 to prove this result. Let S=F1∩F2,
then 0 ≤ |S| ≤ 3(n-2). We will show that, deleting S from
Xn, the subgraph CF1ΔF2,S containing F1ΔF2 has "many"
vertices having degree 3 or more. More precisely, we are going to prove that, in the subgraph CF1ΔF2,S the number of vertices having degree 3 or more is at least 2[3(n-2)+1-|S|]+1 = 6n-2|S|-9. In the following proof, we consider three cases by the size of S: 1) 0 ≤ |S| ≤ n-1, 2) |S|=n, and 3) n+1 ≤ |S| ≤ 3(n-2).
Case 1: 0 ≤ |S| ≤ n-1
Since the connectivity of Xn is n [4], Xn-S is
connected, the subgraph CF1ΔF2,S is the only component
in Xn-S. Since the BC graph Xn has no cycle of length
three and any two vertices have at most two common neighbors, it is straightforward, though tedious, to check that the number of vertices which has degree 2 or 1 is at most 2 in CF1ΔF2,S. Hence, the number of vertices having
degree 3 or more is at least 2n-|S|-2 which is greater than 6n-2|S|-9, for n ≥ 5. By Theorem 2, (F1,F2) is a
distinguishable conditional-pair under the comparison diagnosis model.
Case 2: |S|=n
If Xn-S is disconnected, by Lemma 3, Xn-S has one
trivial component {v} such that N(v)⊂F1 and N(v)⊂F2.
Since F1 and F2 are two conditional faulty sets, this is an
impossible case. So Xn-S is connected, and the subgraph
CF1ΔF2,S is the only component in Xn-S. Let U=Xn-(F1∪F2).
If there exist two vertices u and v in V(U) such that u is adjacent to v, then the condition 1 of Theorem 1 holds and therefore (F1,F2) is a distinguishable conditional-pair;
otherwise V(U) is an independent set. Hence,
NXn-S(v)⊂F1ΔF2, ∀v∈U, and we have the following
inequality 1 2 | degXn S( ) | | degXn S( ) |. v U v F F v v − − ∈ ∈ Δ ≤
∑
∑
To check the inequality, we have
2 | deg ( ) | [2 2(3( 2) 1) | |] | | 2 6 10 n X S v U n n v n S n S n n n n − ∈ ≥ − − + + − = − +
∑
and1 2 2 | deg ( ) | 2[3( 2) 1 | |] 4 10 . n X S v F F v n S n n n − ∈ Δ ≤ − + − = −
∑
2 2 2n 6 10 4 10 for 5, n − n + n> n − n n≥ a contradiction. Case 3: n+1 ≤ |S| ≤ 3(n-2)By Corollary 2, there are four cases in Xn-S we need
to consider. For case 1 of Corollary 2, Xn-S is connected,
the proof is exactly the same as that of Case 2, and hence the detail is omitted. For case 2 and 4 of Corollary 2, Xn-S has at least one trivial component {v} such that N(v)⊂F1 and N(v)⊂F2. Since F1 and F2 are two
conditional faulty sets, the two cases are disregarded. Therefore, we only need to consider that Xn-S has two
components, one of which is K2 and the other one has
2n-|S|-2 vertices. Let (x,y) be the component with only one edge. Since N({x,y})⊆S and F1 and F2 do not
contain all the neighbors of any vertex, vertex x and y cannot belong to F1ΔF2. So the subgraph CF1ΔF2,S is the
other large connected component of Xn-S. Let U=Xn-(F1∪F2)-{x,y}. If there exist two vertices u and v in
V(U) such that u is adjacent to v, then the condition 1 of
Theorem 1 holds and therefore (F1,F2) is a
distinguishable conditional-pair; otherwise V(U) is an independent set. Hence, NXn-S(v)⊂F1ΔF2, ∀v∈U, and we
have the following inequality
1 2 | degXn S( ) | | degXn S( ) |. v U v F F v v − − ∈ ∈ Δ ≤
∑
∑
To check the inequality, we have
2 | deg ( ) | [2 2(3( 2) 1) | | 2] | | 2 6 8 n X S v U n n v n S n S n n n n − ∈ ≥ − − + + − − = − +
∑
and 1 2 2 | deg ( ) | 2[3( 2) 1 | |] 4 12 . n X S v F F v n S n n n − ∈ Δ ≤ − + − ≤ −∑
2 2 2n 6 8 4 12 for 5, n − n + n> n − n n≥ a contradiction. In Case 1, we prove that at least one of the conditions of Theorem 1 is satisfied in subgraph CF1ΔF2,S. In Case 2 and 3, the condition 1 of Theorem 1 holds in subgraph CF1ΔF2,S. Therefore, (F1,F2) is a distinguishableconditional-pair under the comparison model. By Lemma 2, tc(Xn) ≤ 3(n-2) + 1, and by Lemma 6, Xn is conditionally (3(n-2)+1)-diagnosable for n ≥ 5. We
now present our main result which can be stated as follows.
Theorem 4. The conditional diagnosability of Xn is tc(Xn)=3(n-2)+1 for n ≥ 5.
Since Qn, CQn, TQn, MQn ∈ Ln, the following
corollary holds.
Corollary 3. tc(Qn) = tc(CQn) = tc(TQn) = tc(MQn) =
3(n-2)+1 for n ≥ 5.
4. Conclusions
In the real world, processors fail independently and with different probabilities. The probability that any faulty set contains all the neighbors of some processor is very small[1],[8], so we are interested in the study of conditional diagnosability. A new diagnosis measure proposed by Lai et al.[5], it restricts that each processor of a system is incident with at least one fault-free processor. In this paper, we use the BC graph as an example and show that the conditional diagnosability of
Xn is 3(n-2)+1 under the comparison model.
Several different fault diagnosis models have gained much attention in the study of fault diagnosis. It is worth to investigate the conditional diagnosability of a system under various models. It is also an attractive work to develop more different measures of diagnosability based on network topology and network reliability.
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